1. Field of the Invention
The present invention relates to a flatness measuring and analyzing method and, in particular, to a flatness measuring and analyzing method in which a surface form of a plane used as a reference surface of an interferometer or the like is determined by so-called the three-flat method.
2. Description of the Prior Art
Known as a technique for measuring a flatness of an object surface is measurement by an interferometer such as Fizeau interferometer. Though such an interferometer can measure flatness of a sample surface with a high accuracy, this measurement is not absolute measurement but relative measurement with respect to a reference surface. Accordingly, a plane with a very high accuracy is necessary as its reference surface, thereby requiring a method of measuring a reference surface having such a plane of high accuracy.
Known as a technique for measuring such a reference surface is so-called the three-flat method comprising the steps of preparing three sheets of reference plates, measuring the difference between two reference surfaces in each combination of three pairs of reference plates selected from these three plates, and solving simultaneous equations according to the results of measurement, thereby measuring the form of each reference surface.
In the following, this three-flat method will be explained.
Three sheets of reference sheet glass are referred to as A, B, and C. When a coordinate system shown in FIG. 1 is set for each reference glass sheet, forms of the glass surfaces of A, B, and C can be expressed by functions of x and y, whereby they are defined as A (x, y), B (x, y), and C (x, y). Z coordinate shown in FIG. 1 is set in order to specify a glass surface as plus and minus respectively when it is convex and concave.
Here, for example, a reference surface and a sample surface are respectively assumed to be A(x, y) and B(x, y), and these two surfaces are opposed to each other as shown in FIG. 8A so as to be set at a predetermined position of a Fizeau interferometer.
Assuming that the difference between both surfaces measured by this interferometer is .phi.A B(x, y), EQU .phi.AB(x, y)=A(x, y)+B(x', y').
When the coordinates of the sample surface are expressed by the coordinates of the reference surface, B(x', y') is replaced by B(x, -y), thereby EQU .phi.AB(x, y)=A(x, y)+B(x, -y).
Similarly, with respect to other combinations such as those shown in FIGS. 8(b) and 8(c), EQU .phi.CA(x, y)=C(x, y)+A(x, -y) EQU .phi.BC(x, y)=B(x, y)+C(x, -y).
With respect to a line of y=0, EQU .phi.AB(x, 0)=A(x, 0)+B(x, 0) EQU .phi.CA(x, 0)=C(x, 0)+A(x, 0) EQU .phi.BC(x, 0)=B(x, 0)+C(x, 0).
Since .phi.A B(x, 0), .phi.C A(x, 0), and .phi.B C(x, 0) are determined by actual measurement, each form of A(x, y), B(x, y), and C(x, y) is determined when simultaneous equations concerning these three relational expressions are solved.
However, the form determined by the foregoing technique is not a surface form but a cross-sectional form of one line. When the reference surface is rotationally symmetrical, the whole surface form can be recognized by determining a cross-sectional form of one line. Nevertheless, since the reference surface is not rotationally symmetrical in general, a technique for specifying the whole surface form with a high accuracy is needed.
Therefore, the inventors have proposed a method comprising the steps of repeating measurement while relatively rotating the reference surface and the sample surface little by little, simulatively constructing a rotationally symmetrical form by averaging the resulting data, and executing the above-mentioned the three-flat method.
Nevertheless, when such a method is used, the number of measuring operations increases, whereby each operation of measurement and analysis becomes complicated, and errors in measurement may be accumulated.